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A
lvaro Romero-Calvo
Space Propulsion Laboratory,
Department of Aerospace Science and
Technology,
Politecnico di Milano,
Via Giuseppe La Masa, 34,
Milan 20156, Italy
e-mail: alvaro1.romero@mail.polimi.it
Gabriel Cano Gomez
Departamento de Fı
sica Aplicada III,
Universidad de Sevilla,
Avenida de los Descubrimientos s/n,
Sevilla 41092, Spain
e-mail: gabriel@us.es
Elena Castro-Hernandez
Area de Mecanica de Fluidos,
Dep. Ingenierı
a Aeroespacial y Mecanica de
Fluidos,
Universidad de Sevilla,
Avenida de los Descubrimientos s/n,
Sevilla 41092, Spain
e-mail: elenacastro@us.es
Filippo Maggi
Space Propulsion Laboratory,
Department of Aerospace Science and
Technology,
Politecnico di Milano,
Via Giuseppe La Masa, 34,
Milan 20156, Italy
e-mail: filippo.maggi@polimi.it
Free and Forced Oscillations of
Magnetic Liquids Under Low-
Gravity Conditions
The sloshing of liquids in microgravity is a relevant problem of applied mechanics with
important implications for spacecraft design. A magnetic settling force may be used to
avoid the highly non-linear dynamics that characterize these systems. However, this
approach is still largely unexplored. This paper presents a quasi-analytical low-gravity
sloshing model for magnetic liquids under the action of external inhomogeneous magnetic
fields. The problems of free and forced oscillations are solved for axisymmetric geometries
and loads by employing a linearized formulation. The model may be of particular interest
for the development of magnetic sloshing damping devices in space, whose behavior can be
easily predicted and quantified with standard mechanical analogies.
[DOI: 10.1115/1.4045620]
Keywords: computational mechanics, liquid sloshing, ferrohydrodynamics, microgravity
1 Introduction
The term sloshing refers to the forced movement of liquids in par-
tially filled tanks [1]. Propellant sloshing has been a major concern
for space engineers since the beginning of the space era. During
launch, it can result in the partial or total loss of control of the space-
craft [2]. In a low-gravity environment, the liquid tends to adopt a
random position inside the tank and mixes with pressurizing gas
bubbles. This results in a complicated propellant management
system design, often increasing the inert mass of the vehicle [1].
Low-gravity sloshing is characterized by the dominant role of
surface tension that produces a curved equilibrium free surface
(or meniscus) and a complex interaction with the walls of the
vessel that contains the liquid. The first solution of the low-gravity
free surface oscillation problem was given in 1964 by Satterlee and
Reynolds for cylindrical containers [3]. In the context of the Space
Race, a significant effort was made to study low-gravity sloshing in
cylindrical [3–9], spheroidal [8,10,11], or axisymmetric [12–15]
tanks. A non-extensive list of modern works includes numerical
models for cryogens [16,17], coupled non-linear implementations
[18], or computational fluid dynamics (CFD) simulations [19]. Ana-
lytical solutions of the free and forced oscillations problem were
found by Utsumi [20–23].
Different active and passive strategies have been traditionally
employed to mitigate liquid sloshing in microgravity. Active
approaches settle the propellant by imposing an adequate inertial
force with a set of thrusters. Passive techniques make use of
surface tension or membranes to hold the liquid at a certain position
and reduce the effect of random accelerations. The resulting techni-
cal implementations, named Propellant Management Devices, are
currently used to grant adequate liquid propellant feeding in case
of in-orbit ignition of chemical propulsion units [24].
Since the absence of a settling volume force is the main character-
istic of low-gravity sloshing, the problem could be attacked by repro-
ducing the force of gravity with electromagnetic fields if the liquid
can answer to such stimulus. The use of dielectrophoresis, a phenom-
enon on which a force is exerted on dielectric particles in the presence
of a non-uniform electric field, was explored by the US Air Force
with dielectric propellants in 1963 [25]. The study unveiled a high
risk of arcing inside the tanks and highlighted the need for large,
heavy, and noisy power sources. Approaches exploring Magnetic
Positive Positioning have also been suggested to exploit the inherent
magnetic properties of paramagnetic (oxygen) and diamagnetic
(hydrogen) liquids. Relatively recent studies employed numerical
simulations and microgravity experiments to validate this concept
[26,27].
Ferrofluids are colloidal suspensions of magnetic nanoparticles
treated with a surfactant to prevent from agglomeration. As a
result, they exhibit high magnetic susceptibility. Their invention is
attributed to Steve Papell, who in 1963 proposed to “provide an arti-
ficially imposed gravity environment”with ferrofluid-based mag-
netic propellants [28]. The basic equations governing the dynamics
of ferrofluids were presented in 1964 by Neuringer and Rosensweig
[29], giving rise to the field of Ferrohydrodynamics [30]. Although
since then ferrofluids have found numerous applications on Earth,
works addressing their original purpose are scarce. A rare exception
is the NASA Magnetically Actuated Propellant Orientation experi-
ment, which studied the magnetic positioning of liquid oxygen and
validated a custom CFD model with a series of parabolic flight
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received August 26, 2019; final manu-
script received December 3, 2019; published online December 6, 2019. Assoc.
Editor: N. R. Aluru.
Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-1Copyright © 2019 by ASME
experiments with ferrofluids [31]. Subsequent publications pre-
sented numerical models to study and generalize the measurements
for space applications [32–34].
One of the main drawbacks of magnetic sloshing control is the
rapid decay of magnetic fields, which limits its applicability to
small and compact tanks. In this context, the increasing number of
propelled microsatellites may benefit from this technology since a
direct control of liquid sloshing could be achieved with small
low-cost magnets. Once the liquid is positioned, magnetic fields
could also be used to tune the natural frequencies and damping
ratios of the system. This approach has been adopted for terrestrial
applications, such as tuned magnetic liquid dampers [35,36].
On-ground research exploring axisymmetric sloshing [37,38], the
frequency shifts due to the magnetic interaction [39], two-layer
sloshing [40], or the swirling phenomenon [41], among others, has
been carried out in the past with notable results. The sloshing of fer-
rofluids in low-gravity was indirectly studied in 1972 with a focus on
gravity compensation techniques [42].
This work addresses the free and forced oscillations of magnetic
liquids in axisymmetric containers when subjected to an external
inhomogeneous magnetic field in microgravity. A ferrohydrody-
namic model is developed to predict the natural frequencies and
modal shapes of the system and a case of application is presented.
Unlike non-magnetic low-gravity sloshing, the presence of a restor-
ing force ensures that the hypothesis of small oscillations (linear
sloshing) is satisfied in a wide range of operations.
2 Problem Formulation
The system to be modeled is represented in Fig. 1. A volume Vof a
magnetic liquid fills an upright axisymmetric tank with radius aat the
meniscus contour. The liquid is incompressible, Newtonian and is
characterized by a density ρ, specific volume v=ρ
−1
, kinematic vis-
cosity ν, surface tension σ, and magnetization curve M(H). Hand M
are, respectively, the modules of the magnetic field Hand magneti-
zation field M, which are assumed to be collinear. The liquid
meets the container wall with a contact angle θ
c
. An applied inhomo-
geneous axisymmetric magnetic field H
0
is imposed by an external
source (e.g., a coil located at the base of the container). The inertial
acceleration galong the z-axis is also considered. A non-reactive
gas at pressure p
g
fills the free space. In the figure, sis a curvilinear
coordinate along the meniscus with origin in the vertex Oand the rel-
ative heights are given by w(fluid surface–vertex), f(meniscus–
vertex) and h(fluid surface–meniscus). The container is subjected
to a lateral displacement x(t). The meniscus is represented by a
dashed line, and the dynamic fluid surface is given by a solid line.
The model here presented extends the works by Satterlee and
Reynolds [3] and Yeh [12] by considering the magnetic interaction
and the axisymmetric oscillations case.
2.1 Nonlinear Formulation. A cylindrical reference system
{u
r
,u
θ
,u
z
}, centered at the vertex of the meniscus, is subsequently
considered. If an irrotational flow field is assumed, there exists a
potential φsuch that
v=−∇φ=−φrur−1
rφθuθ−φzuz(1)
being vthe flow velocity with the subindices denoting the deriva-
tive. The velocity potential satisfies Laplace’s equation
∇2φ=φrr +φr
r+φθθ
r2+φzz =0inV(2)
subjected to the non-penetration wall boundary condition
φr=−˙
xcos θ,φθ/r=˙
xsin θ,φz=0onW(3)
An additional boundary condition at the free surface is given by the
unsteady ferrohydrodynamic Bernoulli’s equation, which for an iso-
thermal system with collinear magnetization field Madopts the
form [30,43]
−˙
φ+v2
2+p*
ρ+gw −ψ
ρ+˙
xcos θφr−˙
xsin θφθ
r
=β(t)onS(4)
where gis the inertial acceleration, ψis the magnetic force potential,
β(t) is an arbitrary function of time, and p* is the composite pres-
sure,defined as [30]
p*=p(ρ,T)+μ0H
0
v∂M
∂v
H,T
dH+μ0H
0
M(H)dH(5)
with the first, second, and third terms being named thermodynamic,
magnetostrictive, and fluid-magnetic pressure, respectively. For
magnetically diluted systems M∼ρ, where ρis the concentration
of magnetic particles for the case of ferrofluids. Under this addi-
tional assumption, both pressure-like components are approxi-
mately compensated, and hence p*≈p(ρ,T)[30].
The canonical magnetic force per unit volume is given by
μ0M∇H, with μ
0
=4π·10
−7
N/A
2
being the permeability of free
space [30]. It can be shown that, for an isothermal fluid, this
force derives from the potential [29]
ψ=μ0H
0
M(H)dH(6)
Due to the discontinuity of the Maxwell stress tensor at the mag-
netic liquid interface, the ferrohydrodynamic boundary condition
in the absence of viscous forces becomes
p*=pg−pc−pnon S(7)
where pn=μ0M2
n/2 is the magnetic normal traction,M
n
is the mag-
netization component normal to the fluid surface, and p
c
is the cap-
illary pressure. The last is defined by the Laplace-Young equation
p
c
=σK, where
K=1
r
∂
∂r
rwr
1+w2
r+1
r2w2
θ
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦
+1
r2
∂
∂θ
wθ
1+w2
r+1
r2w2
θ
⎡
⎢
⎢
⎣⎤
⎥
⎥
⎦
(8)
is the curvature of the surface [1]. Since at Eq. (4) only the spatial
derivatives of the velocity potential have a physical meaning (e.g.,
Fig. 1 Geometry of the system under analysis, composed of a
magnetic liquid that fills a container in microgravity while sub-
jected to an external magnetic field. S′and C′refer to the menis-
cus surface and contour, while S and C are the dynamic fluid
surface and contour, respectively. O denotes the vertex of the
meniscus, W is the vessel wall, and V denotes the fluid volume.
021010-2 / Vol. 87, FEBRUARY 2020 Transactions of the ASME
Eq. (1)), any function of time can be added to φif mathematically
convenient. From a physical viewpoint, the absolute value of p
remains undetermined under the incompressible flow assumption
[43]. The integration constant β(t) can be then absorbed into the def-
inition of φ. By arbitrarily selecting β(t)=p
g
/ρ, the dynamic inter-
face condition is obtained
˙
φ−1
2φ2
r+1
r2φ2
θ+φ2
z
+σ
ρK−gw +ψ
ρ+μ0M2
n
2ρ
−˙
xcos θφr+˙
xsin θφθ
r=0onS
(9)
In an inertial reference system, the vertical displacement wof a
surface point lying at (r,θ) in the interface z=w(r,θ,t) is given by
dw
dt=˙
w+wr
dr
dt+wθ
dθ
dton S(10)
If the velocity components relative to the tank dw/dt,dr/dt, and
rdθ/dtare expressed as a function of the potential given by
Eq. (1), the kinematic interface condition that relates the last with
the shape of the free surface is
˙w=−φz+wrφr+˙xcos θ
+wθ
r2φθ−˙xr sin θ
on S(11)
The continuity equation given by Eq. (2), the kinematic relation in
Eq. (11) and the boundary conditions in Eqs. (3) and (9),define the
problem to be solved after imposing the contact angle at the wall
(θ
c
) and a contact hysteresis parameter that will be described later
in the text.
2.2 Equilibrium Free Surface Shape. Due to the axisymme-
try of geometry and loads, the static equilibrium surface of the fluid
(S′) is also axisymmetric. Its shape can be determined from the
balance of vertical forces in a circular crown of inner radius rand
infinitesimal width dr along the surface [24]. This results in the fol-
lowing set of dimensionless differential equations:
d
dSRdF
dS
=RdR
dSλ+BoF −ψ(R)
(12a)
dF
dS
d2F
dS2+dR
dS
d2R
dS2=0(12b)
and boundary conditions
R(0) =F(0) =dF(0)
dS=0,dR(0)
dS=1(12c)
dF(1)
dR=tan π
2−θc
(12d)
where R=r/a,F=f/a,S=s/a,Bo =ρga
2
/σis the Bond number,
λ=a(p
g
−p
0
)/σ, being p
0
the liquid pressure at the free surface
vertex, ψincludes the magnetic potential and magnetic normal trac-
tion through
ψ(R)=aμ0
σH(R,F(R))
H(0,0)
M(H)dH+M2
n
2
F(R)
(13)
and the static contact angle with respect to the vertical θcis given by
θc=θc+π
2−arctan dW
drC′
(14)
A numerical solution can be easily computed by (1) setting an initial
vertex position, (2) calculating the value of λiteratively in order to
satisfy the contact angle condition given by Eq. (12d), (3) solving
the system with an ODE solver, and (4) obtaining the new height
of the vertex through volume conservation. The procedure is
repeated until the vertex height converges with a prescribed relative
variation. When non-trivial magnetic setups are involved, a FEM
simulation must be included in the loop.
2.3 Linear Equations. The dynamic and kinematic conditions
in Eqs. (9) and (11) are highly nonlinear. The standard analytical
procedure overcomes this difficulty by linearizing the problem
and restricting the analysis to small oscillations. If the wave position
is expressed as the sum of the static equilibrium solution and a small
perturbation
w(r,θ,t)=f(r)+h(r,θ,t)(15)
it will be possible to express the system of equations and boundary
conditions as a Taylor’s series expansion around the equilibrium
surface S′. If second-order terms are neglected, the boundary-value
problem becomes
∇2φ=0inV(16a)
φr=−˙
xcos θ,φθ/r=˙
xsin θ,φz=0onW
˙
φ+σ
ρ
1
r
∂
∂r
rhr
1+f2
r
3/2
+1
r2
∂
∂θ
hθ
1+f2
r
−g−μ0
ρM∂H
∂z+Mn
∂Mn
∂z
h=0onS′
(16b)
˙
h=−φz+fr(φr+˙
xcos θ)onS′(16c)
hr=γhon C′(16d)
Equation (16d)assumes that the slope of the perturbation field at
the wall is related to the magnitude of the perturbation at the same
point through the parameter γ. The free-edge condition is character-
ized by γ=0, while the stuck-edge condition is characterized by
γ→∞[12]. This assumption is far from being rock-solid and has
indeed motivated a strong debate in the past. It has been suggested
that the contact angle hysteresis condition depends not only on the
position of the wave but also on its velocity [4] or the state of the
wall [7]. In the absence of a clear criteria, some studies assume
the free-edge condition or intermediate approaches, generally
obtaining a reasonable agreement with experimental data [10].
The only difference between the previous formulation and the
classical problem without magnetic interactions is given by the
magnetic term in Eq. (16b). The effective gravity acceleration
includes both inertial and magnetic components and is given by
g*(r)=g−μ0
ρM∂H
∂z+Mn
∂Mn
∂z
S
(17)
where it can be observed that the magnetic contribution at the
surface is a function of the radius. The magnitude and relative
importance of the magnetic terms depend on the magnetic configu-
ration and gravity level of the system under analysis. In particular,
the magnetic component will be more significant in the absence of
gravity.
The magnetic field modifies the effective gravity acceleration of
the system and shifts its natural frequencies, as reported in normal-
gravity works [35,44]. If the magnetic term was approximately
constant in R, like in the case of a linear magnetic field and a flat
surface, the problem would be equivalent to the non-magnetic
system [42]. In this analysis, however, an inhomogeneous magnetic
field is considered.
Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-3
2.4 Extraction of Tank Motion. The potential φhas been
referred to an inertial reference system. In order to analyze the
movement of the free surface in the tank reference frame, it
would be convenient to split this potential into rigid body (ϕ
0
)
and perturbed (ϕ) components, so that
φ=ϕ0+ϕ,ϕ0=−˙
xrcos θ(18)
The boundary-value problem can be then expressed as a function of
the perturbed potential and becomes
∇2ϕ=0inV(19a)
ϕn=0onW(19b)
˙
ϕ+σ
ρ
1
r
∂
∂r
rhr
1+f2
r
3/2
+1
r2
∂
∂θ
hθ
1+f2
r
−g−μ0
ρM∂H
∂z+Mn
∂Mn
∂z
h=¨
xrcos θon S′
(19c)
˙
h=−ϕz+ϕrfron S′(19d)
hr=γhon C′(19e)
3 Free Oscillations Problem
3.1 Dimensionless Linear Equations. In Refs. [3,12], it is
proposed to split the potentials ϕand hinto spatial and temporary
components, the second being a cyclic function of time with a cir-
cular frequency ω. The resulting dimensionless boundary-value
problem is
∇2Φ=0 inV (20a)
∂Φ
∂n=0onW(20b)
Ω2Φ−[Bo +Bomag(R)]H
+1
R
∂
∂R
RHR
1+F2
R
3/2
+1
R2
∂
∂θ
Hθ
1+F2
R
=0onS
′
(20c)
H=ΦZ−ΦRFRon S′(20d)
HR=ΓHon C′(20e)
where R=r/a,Z=z/a,F=f/a,ϕ(R,θ,Z,t)=
g0a3
Φ(R,θ,Z)
sin(ωt), h(R,θ,t)=
ag0/ω2
H(R,θ) cos(ωt), Ω
2
=ρa
3
ω
2
/σ,Γ=
aγ, and g
0
is the acceleration of gravity at ground level [1]. The
Magnetic Bond Number has been defined as
Bomag(R)=−μ0a2
σM∂H
∂z+Mn
∂Mn
∂z
F(R)
(21)
and accounts for the effects of the external magnetic field on the
liquid.
3.2 Variational Formulation. By following the procedure
described in Refs. [3,12], it is possible to develop a variational prin-
ciple equivalent to Eqs. (20b)and (20c)as
I=S′
H2
R
1+F2
R
3/2+1
R2
H2
θ
1+F2
R
1/2
+Bo +Bomag(R)
H2−Ω2ΦHRdRdθ
−Ω2W
ΦGR dRdθ−ΓC′
H2
1+F2
R
3/2
R=1
dθ
=extremum
(22a)
subjected to
∇2Φ=0inV(22b)
H=ΦZ−FRΦRon S′(22c)
G=ΦZ−WRΦRon W (22d)
HR=ΓHon C′(22e)
where Gand its associated terms arise from the application of the
wall boundary condition given by Eq. (20b)as detailed in
Ref. [12]. The obtention of this variational formulation follows
the procedure described in Refs. [12,45].
3.3 Ritz Method. The previous set of equations can only be
analytically solved for simplified configurations in the absence of
magnetic fields, like the case of a cylindrical container with a flat
bottom and flat fluid surface (θc=90 deg) [3]. For other physical
systems, Ritz approximations [1,12]orfinite differences approaches
[5,6] have been proposed to compute the eigenfunctions of the
problem. The basic formulation of the first approach is subsequently
developed based on Refs. [3,12].
By following Ritz’s method, the eigenfunctions Φ
(n)
can be
approximated as the linear combination of admissible functions
Φi(R,θ,Z) that satisfy the boundary conditions of the problem
described by Eqs. (22b)–(22e). This results in
Φ(n)=
N
i=1
C(n)
iΦi(n=1,...,N)(23)
where Nis the size of the set of admissible functions. In the same
way, the eigenfunctions H(n)
iand G
(n)
are approximated by
ζi(R,θ) and ξi(R,θ) through
H(n)=
N
i=1
C(n)
iζi(24)
G(n)=
N
i=1
C(n)
iξi(25)
The sets of admissible functions are linked through Eqs. (22c)and
(22d).IfΦ
(n)
,H(n), and G
(n)
are continuous functions of C(n)
i, the
extremum condition represented by Eq. (22a)requires that
∂I
∂C(n)
i
=0,(i=1,2,...,N)(26)
which results in the system of equations
N
i=1
C(n)
iRij +BoLij +Lmag
ij −Ω2
nQij
=0,
(j=1,2,...,N)
(27)
021010-4 / Vol. 87, FEBRUARY 2020 Transactions of the ASME
being
Rij =F
ζiRζjR
(1 +F2
R)3/2+n2ζiζj
R2(1 +F2
R)1/2
RdRdθ
−Γ2π
0
ζiζj
(1 +F2
R)3/2
R=1
dθ
(28a)
Lij =F
ζiζjRdRdθ(28b)
Lmag
ij =F
Bomag(R)ζiζjRdRdθ(28c)
Qij =1
2F
Φiζj+Φjζi
RdRdθ
+1
2F
Φiξj+Φjξi
RdRdθ
(28d)
The system has a nontrivial solution only when its determinant is
zero. The eigenvalues Ω2
n, and therefore the corresponding modal
circular frequencies ω
n
, are then computed by means of the charac-
teristic equation:
Rij +BoLij +Lmag
ij −Ω2Qij=0(29)
Once solved, the eigenfunctions of the problem are obtained from
Eq. (23) to Eq. (25).
3.4 Forced Lateral Oscillations and Mechanical Analogies.
In order to solve the forced lateral oscillations case, a modal solu-
tion for the linearized boundary-value problem given by Eq. (19)
is built from the eigenmodes Φ
(n)
and H(n). This solution is
expressed as
ϕ=
N
n=1
An(t)Φ(n),h=
N
n=1
Bn(t)H(n)(30)
where the modal coordinates A
n
(t) and B
n
(t) are computed from
their corresponding modal equations. The reader is referred to
Ref. [12] for a full description of this procedure.
Since the modal equations are linear, the forced sloshing problem
can be conceived as the superposition of several linear oscillators.
These are usually assumed to be a series of spring-mass systems
on which a linear damper is included a posteriori. The employment
of a mechanical analogy simplifies the integration of the sloshing
problem into the equations of motion of the vehicle. In the case
here analyzed, an additional magnetic pressure should be consid-
ered. However, in virtue of Newton’s action-reaction principle, if
the magnetic source is rigidly coupled to the tank then the distribu-
tion of magnetic pressures cannot produce torque in the assembly.
That is, non-magnetic mechanical analogies can be extended for
the magnetic case by simply employing the new magnetic eigen-
modes Φ
(n)
and H(n)with their corresponding eigenfrequencies.
Some possibilities are the models developed by Dodge and Garza
[46] or Utsumi [23].
3.5 Selection of Admissible and Primitive Functions. The
set of admissible functions for Φ,H, and G, related through
Eqs. (22c)and (22d)satisfy by definition Eqs. (22b)–(22e)and
form truncated series that approximate the eigenfunctions of the
problem. A set of primitives should be previously defined as [3]
ϑp=Jn(kpR) cos(mθ)ekpZ(p=1,...,N,N+1) (31a)
ζp=ϑpZ −FRϑpR
Z=F(R)(31b)
ξp=ϑpZ −WRϑpR
Z=W(R)(31c)
with k
p
being the roots of the equation
d
dRJn(kiR)
R=1
=0(32)
where J
n
is the Bessel function of first kind and order n. This index
is used to study the axisymmetric (n=0) and lateral (n=1) cases,
while mdefines the circumferential symmetry of the problem. Axi-
symmetric primitive functions will be characterized by n=m=0,
while lateral sloshing functions will be characterized by n=m=1.
However, the previous set of primitives does not satisfy
Eq. (22e). The set of admissible functions is then created as a
linear combination of the previous
Φi=
N+1
p=i
aipϑp,ζi=
N+1
p=i
aipζp,ξi=
N+1
p=i
aipξp,(i=1,2,...,N)
(33)
The N+1−pcoefficients a
ip
for each ivalue are determined by
imposing (i) a normalization condition, (ii) a contact angle value,
and (iii) a Lagrange minimization problem designed to produce
Bessel-like functions. These conditions are, respectively, expressed
as [3]
N+1
p=i
aipζp(1) =1(34a)
N+1
p=i
aipζpR (1) =Γ(34b)
N+1
p=i
aip(Kpj −k2
pLpj)+λ1iζjR (1)
+λ2iζj(1) =0(j=i,i+1,...,N,N+1)
(34c)
where λ
1i
and λ
2i
are the Lagrange multipliers of the minimization
problem and
ζi(R)=ζi(R,θ)/cos(mθ)(35)
Kij =F
ζiRζjR −n
R2ζiζj
RdRdθ(36)
Lij =F
ζiζjRdRdθ(37)
Once the system is solved, the admissible set can be used to solve
the eigenvalue problem.
The success of this method depends on finding an adequate set of
admissible functions Φisuch that the eigenfunctions Φ
(n)
can be
represented with a reduced number of elements. The Zterm in the
primitives ϑ
p
, evaluated at the equilibrium surface, grows exponen-
tially when F(R) departs significantly from Z=0. This is the case of
low Bond numbers and small contact angles. In Ref. [13], it is stated
that for contact angles lower than 15 deg in the case of free-edge
condition (Γ=0) or lower than 60 deg for the stuck-edge condition
(Γ→∞), the system may become numerically ill-conditioned. Fur-
thermore, the comparison between this method and a finite differ-
ences approach showed significant divergences in the shape of
the eigenfunctions Φ
(n)
for particular cases.
A potential solution would be finding a set of primitive functions
without exponential terms. To the best knowledge of the authors
Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-5
and considering the attempts made in Ref. [3], an alternative has not
yet been proposed. It should also be noted that the magnetic force
generally flattens the equilibrium surface, hence mitigating the
effect of the exponential term in Eq. (31a).
4 Case of Application
4.1 System Description. CubeSats are a particular class of
nanosatellites composed of standardized 10 × 10 cm cubic units
(U) [47]. Their subsystems are designed to fit this standard, lower-
ing the manufacturing costs and enhancing adaptability (e.g., the
Aerojet MPS-130 propulsion module is offered in 1U or 2U
formats). Due to the rapid decay of magnetic fields with distance,
small spacecrafts with small propellant tanks may be particularly
well suited for magnetic sloshing control implementations. Their
development requires a dedicated feasibility study that is beyond
the scope of this paper; however, the performance of a hypothetical
system is here addressed.
A 1U cylindrical container with 10 cm height, 5 cm radius, and
filled with a ferrofluid solution up to half of its height is subse-
quently considered. In order to generate a downward restoring
force, a high-end cylindrical neodymium magnet magnetized at
1500 kA/m in the vertical direction is placed under the tank. The
magnet has a hole of r
i
=2.5 mm radius at the center (liquid
outlet), a height h
m
, an external radius r
e
, and a density ρ
m
=
7010 kg/m
3
. A contact angle of 67 deg is assumed in microgravity
conditions. The sketch of this setup is given in Fig. 2.
The liquid is a 1:10 volume solution of the Ferrotec water-based
EMG-700 ferrofluid with a 0.58% volume concentration of mag-
netic nanoparticles. Its magnetic properties were measured with a
MicroSense EZ-9 Vibrating Sample Magnetometer at the Physics
Department of Politecnico di Milano. The corresponding magneti-
zation curve is depicted in Fig. 3and shows an initial susceptibility
χ=0.181 and saturation magnetization M
s
=3160 A/m. Viscosity
is assumed to have a negligible effect on the free sloshing problem.
4.2 Magnetic Modeling. The magnetic system is modeled in
COMSOL MULTIPHYSICS, which is interfaced with the model developed
in Sec. 2. The FEM simulation is employed to estimate the fields H
and Mfor a given magnet and equilibrium surface shape (menis-
cus). The last is computed iteratively by means of Eq. (12) with a
FEM-in-the-loop implementation. Equation (21) is then employed
to calculate the magnetic Bond number at the surface,
which determines the solution of the free oscillation problem. The
eigenfrequencies and eigenmodes of the system are finally obtained
by solving Eq. (22).
To simulate the magnetic field, the model solves the stationary
non-electric Maxwell equations
∇×H=0(38)
B=∇×A(39)
where Ais the magnetic vector potential produced by the magne-
tized materials. The constitutive relation
B=μ0H+M() (40)
is applied to the magnet with M=[0, 0, 1500] kA/m and to the sur-
rounding air with M=0. The magnetization curve M=f(H)in
Fig. 3is applied to the ferrofluid volume.
The simulation domain is a rectangular 1 × 2 m region enclosing
the container. An axisymmetric boundary condition is applied to the
symmetry axis, while the tangential magnetic potential is imposed
at the external faces through n×A=n×A
0
.A
0
is the dipole term
of the magnetic vector potential generated by the magnetization
fields of the magnet and ferrofluid. Consequently, A
0
is computed
as the potential vector generated by two point dipoles applied at
the centers of the magnetization distributions and whose dipole
moments are those of said distributions. While the dipole associated
with the magnet can be calculated beforehand, the ferrofluid dipole
needs to be approximated iteratively by integrating Min the ferro-
fluid volume. The relative error in the magnetic vector potential due
the dipole approximation is estimated to be below 0.03% at the
boundary of the domain with respect to the exact value generated
by the equivalent circular loop.
The mesh is composed of 53,000 irregular triangular elements
and is refined at the meniscus, as shown in Fig. 4. Mean and
minimum condition numbers of 0.983 and 0.766 are measured.
Figure 5shows a particular configuration of analysis. Positive
and negative curvatures are observed at the meniscus due to the
high intensity of the magnetic field. Weaker magnets would result
in convex equilibrium surfaces, as in the non-magnetic case. It
should be noted that the magnetic Bond number rapidly decreases
with distance to the source, spanning between 0 and 20 at the free
surface.
4.3 Parametric Analysis. Figure 6depicts the fundamental
sloshing frequency ω
1
, corresponding to the lowest root of
Eq. (29), as a function of the external radius r
e
and height h
m
of
the magnet. N=7 admissible functions were employed in the com-
putation. The mass of the magnet is given in a second scale, reflect-
ing the technical trade-off between mass and sloshing frequency.
Larger magnets result in stronger restoring forces and higher
Fig. 2 Sketch of the case of application. Units in millimeters.
Fig. 3 Magnetization curve of the 1:10 solution of the Ferrotec
EMG-700 water-based ferrofluid
021010-6 / Vol. 87, FEBRUARY 2020 Transactions of the ASME
sloshing frequencies. For example, a downward 7 N force and a
100% increase in the fundamental sloshing frequency can be
achieved with a 3 mm height, 30 mm external radius, and 60 g
magnet. Unlike the non-magnetic case, the presence of a significant
restoring force ensures that the assumption of small oscillations
(linear sloshing) is not violated for moderate displacements of the
container.
A small fluctuation in the frequency plot appears due to numeri-
cal errors. The procedures for solving Eq. (12) are highly dependent
on the initial estimation of λ. This behavior has been extensively
reported in the bibliography [3,24] and is further complicated by
the magnetic interaction. In addition, the eigenvalue problem that
provides the natural frequencies of the system becomes ill-
conditioned if the actual modal shapes diverge significantly from
the primitive functions [13]. For strong magnetic fields, this may
certainly be the case. The problem would be solved if a finite differ-
ences approach is employed instead of Ritz’s method.
The modal shapes for the 3 mm height and 30mm external radius
magnet are represented in Fig. 7together with the non-magnetic
modes (i.e., the ones obtained when the magnet is removed).
Although the profiles are essentially the same, it is interesting to
observe how the fundamental mode slightly reduces and increases
its vertical displacement where the magnetic Bond number is
greater and smaller, respectively. This is consistent with the afore-
mentioned stabilizing role of the magnetic force.
5 Conclusions
A quasi-analytical model has been developed to study the slosh-
ing of magnetic liquids in low-gravity conditions. The magnetic
interaction modifies the shape of the meniscus and the effective
inertial acceleration of the system as shown in Eqs. (13) and (17),
Fig. 4 Mesh and overall dimensions of the magnetic FEM simu-
lation domain. Units in millimeters.
Fig. 5 Equilibrium configuration of the magnetic sloshing
damping system for h
m
=5 mm and r
e
=2.8 cm. The magnetic
Bond number is represented in the color scale for the range of
interest.
Fig. 6 Fundamental sloshing frequency ω
1
(top) and mass of the
magnet (bottom) as a function of the height h
m
and external
radius r
e
of the magnet
Fig. 7 Magnetic and non-magnetic sloshing modes shape for
the case under analysis
Journal of Applied Mechanics FEBRUARY 2020, Vol. 87 / 021010-7
respectively. As a consequence, a shift of the eigenfrequencies and a
modification of the eigenmodes of the free oscillations problem is
produced. The framework here presented extends the models devel-
oped in previous works [3,12] by adding the magnetic and axisym-
metric cases.
The small oscillations assumption is generally not valid for non-
magnetic sloshing in microgravity, which is usually characterized
by complex non-linear deformations driven by surface tension. In
the magnetic case, however, the linear treatment of the problem is
endorsed by a significant magnetic restoring force. It would be rea-
sonable to ask how strong magnetic fields affect the shape of the
eigenmodes and hence the reliability of Ritz’s approximation.
Since this procedure has been historically discussed [13], the devel-
opment of a finite differences model becomes particularly conve-
nient for future implementations. Ritz’s method represents,
however, the simplest tool to solve the variational formulation
given by Eq. (22) and has been presented for illustrative purposes.
From a technical perspective, the magnetic sloshing concept rep-
resents an opportunity to develop new sloshing control devices for
microsatellites. Unlike the non-magnetic case, the response of the
system can be easily predicted, quantified, and simulated by
means of standard mechanical analogies. These simplified models
can be easily embedded in a controller (e.g., a linear observer)
used to predict and compensate the sloshing disturbances of a
spacecraft in orbit. The spacecraft would then benefit from a signif-
icantly improved pointing performance.
Acknowledgment
The authors thank their institutions, Politecnico di Milano and the
University of Seville, for their financial and academic support. The
discussions with Prof. Miguel Ángel Herrada Gutiérrez on the ver-
ification of the non-magnetic model were highly appreciated.
Nomenclature
a=axisymmetric tank radius at the contour of the
meniscus
f=relative height between meniscus and vertex
g=inertial acceleration
h=relative height between meniscus and dynamic
liquid surface
p=thermodynamic pressure
s=curvilinear coordinate along the meniscus
v=specific volume
w=relative height between dynamic liquid surface and
vertex
x=lateral displacement of the container
n=unitary external vector normal to the fluid surface
v=liquid velocity
C=dynamic contour
F=dimensionless f
G=wall boundary condition function
I=variational principle
K=curvature of the liquid surface
N=size of the set of admissible functions
O=vertex of the meniscus
S=dynamic fluid surface
V=liquid volume
W=walls of the container
Z=dimensionless z
A=magnetic vector potential
B=magnetic flux density
H=magnetic field
M=magnetization field
H=dimensionless h
S=dimensionless s
a
ip
=modal coefficients used by Φi,ζi, and ξi
g
0
=gravity acceleration at ground level
h
m
=height of the cylindrical magnet
p
0
=thermodynamic pressure at the vertex of the
meniscus
p
c
=capillary pressure
p
g
=filling gas pressure
p
n
=magnetic normal traction
r
e
=external radius of the cylindrical magnet
r
i
=internal radius of the cylindrical magnet
A
n
=forced problem coefficients for ϕ
B
n
=forced problem coefficients for h
M
n
=magnetization component normal to the fluid
surface
Q
ij
=matrix from Eq. (29)
R
ij
=matrix from Eq. (29)
p*=composite pressure
C′=Meniscus contour
S′=Meniscus surface
G
(n)
=eigenfunctions of G
C(n)
i=modal coefficients used by Φ
(n)
,H(n), and G
(n)
Bo =bond number
Bo
mag
=magnetic bond number
w=relative height between dynamic liquid surface and
vertex for a particular surface point
A
0
=dipole term of A
H
0
=applied magnetic field
Lij =matrix from Eq. (29)
Lmag
ij =matrix from Eq. (29)
H(n)=eigenfunctions of H
β=arbitrary time constant of Bernoulli’s equation
Γ=dimensionless γ
γ=surface hysteresis parameter
ζ
p
=primitive functions of ζi
ζi=admissible functions of H(n)
ϑ
p
=primitive functions of Φi
θ
c
=surface contact angle
θc=surface contact angle referred to the vertical
λ=equilibrium free surface parameter
μ
0
=magnetic permeability of free space
ν=kinematic viscosity
ξ
p
=primitive functions of ξi
ξi=admissible functions of G
(n)
ρ=liquid density
σ=surface tension
φ=liquid velocity potential
ϕ
0
=rigid-body liquid velocity potential
ϕ=perturbed liquid velocity potential
Φ=dimensionless ϕ
Φ
(n)
=eigenfunctions of Φ
Φi=admissible functions of Φ
(n)
χ=magnetic susceptibility
ψ=magnetic force potential
ψ=dimensionless magnetic term at the meniscus
Ω=dimensionless ω
Ω
n
=dimensionless ω
n
ω=circular frequency of the surface wave
ω
n
=modal circular frequency of the surface wave
{r,θ,z}=cylindrical coordinates of the system {u
r
,u
θ
,u
z
}
{u
r
,u
θ
,u
z
}=cylindrical reference system centered at the vertex
of the meniscus
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